Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number TheoryPlaylist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
http://www.youtube.com/subscription_center?add_user=SocraticaStudios
Subject: Number Theory
Teacher: Michael HarrisonArtist: Katrina de Dios

published:12 Jan 2012

views:127185

Subscribe Now:
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Watch More:
http://www.youtube.com/Ehow
Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Computer Science on Khan Academy: Learn select topics from computer science - algorithms (how we solve common problems in computer science and measure the efficiency of our solutions), cryptography (how we protect secret information), and information theory (how we encode and compress information).
About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Computer Science channel: https://www.youtube.com/channel/UC8uHgAVBOy5h1fDsjQghWCw?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

published:29 Apr 2014

views:61939

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:19 May 2014

views:24056

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
Subscribe to HipHopDX on Youtube: http://bit.ly/dxsubscribe
Produced, Written & Hosted By: Murs
instagram.com/murs316
twitter.com/murs
twitch.tv/murs
Produced, Shot & Edited By: James Kreisberg
instagram.com/rolltheclipjames
Join the discussion on all socials #DXBreakdown
Check out more of DX here:
http://www.hiphopdx.com
https://twitter.com/hiphopdx
https://www.facebook.com/hiphopdx
http://instagram.com/hiphopdx
For over 17 years, HipHopDX has been at the forefront of Hip Hop culture online, featuring over 2.7 million readers per month. As one of the longest-standing Hip Hop websites, DX not only stays current on Hip Hop culture, but continues to influence it, encourage it, and simultaneously reflect on its past. Our insightful, honest editorials, unbiased reviews, premier audio and video sections, and original video content, draws one of the most loyal followings online. Check us out at hiphopdx.com
-~-~~-~~~-~~-~-
Please watch: "Hopsin Talks End Of Funk Volume, Immaturity & Undercover Prodigy | SoulfulSundays"
https://www.youtube.com/watch?v=3V8Ds3q2tZA
-~-~~-~~~-~~-~-

published:06 Jan 2018

views:118044

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

Khan Academy

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with the aim of providing a free, world-class education for anyone, anywhere. The organization produces short lectures in the form of YouTube videos. In addition to micro lectures, the organization's website features practice exercises and tools for educators. All resources are available for free to anyone around the world. The main language of the website is English, but the content is also available in other languages.

In late 2004, Khan began tutoring his cousin Nadia who needed help with math using Yahoo!'s Doodle notepad.When other relatives and friends sought similar help, he decided that it would be more practical to distribute the tutorials on YouTube. The videos' popularity and the testimonials of appreciative students prompted Khan to quit his job in finance as a hedge fund analyst at Connective Capital Management in 2009, and focus on the tutorials (then released under the moniker "Khan Academy") full-time.

Computer science

Computer science is the scientific and practical approach to computation and its applications. It is the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems.

Theory

Theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking. Depending on the context, the results might for example include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several different related meanings. A theory is not the same as a hypothesis. A theory provides an explanatory framework for some observation, and from the assumptions of the explanation follows a number of possible hypotheses that can be tested in order to provide support for, or challenge, the theory.

A theory can be normative (or prescriptive), meaning a postulation about what ought to be. It provides "goals, norms, and standards". A theory can be a body of knowledge, which may or may not be associated with particular explanatory models. To theorize is to develop this body of knowledge.

As already in Aristotle's definitions, theory is very often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for "doing", which is opposed to theory because pure theory involves no doing apart from itself. A classical example of the distinction between "theoretical" and "practical" uses the discipline of medicine: medical theory involves trying to understand the causes and nature of health and sickness, while the practical side of medicine is trying to make people healthy. These two things are related but can be independent, because it is possible to research health and sickness without curing specific patients, and it is possible to cure a patient without knowing how the cure worked.

Number Theory: Fermat's Little Theorem

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number TheoryPlaylist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
http://www.youtube.com/subscription_center?add_user=SocraticaStudios
Subject: Number Theory
Teacher: Michael HarrisonArtist: Katrina de Dios

1:45

An Introduction to Number Theory : College Math

An Introduction to Number Theory : College Math

An Introduction to Number Theory : College Math

Subscribe Now:
http://www.youtube.com/subscription_center?add_user=Ehow
Watch More:
http://www.youtube.com/Ehow
Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Computer Science on Khan Academy: Learn select topics from computer science - algorithms (how we solve common problems in computer science and measure the efficiency of our solutions), cryptography (how we protect secret information), and information theory (how we encode and compress information).
About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Computer Science channel: https://www.youtube.com/channel/UC8uHgAVBOy5h1fDsjQghWCw?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

48:27

MathHistory22: Algebraic number theory and rings I

MathHistory22: Algebraic number theory and rings I

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

13:00

The Magic Number Theory

The Magic Number Theory

The Magic Number Theory

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
Subscribe to HipHopDX on Youtube: http://bit.ly/dxsubscribe
Produced, Written & Hosted By: Murs
instagram.com/murs316
twitter.com/murs
twitch.tv/murs
Produced, Shot & Edited By: James Kreisberg
instagram.com/rolltheclipjames
Join the discussion on all socials #DXBreakdown
Check out more of DX here:
http://www.hiphopdx.com
https://twitter.com/hiphopdx
https://www.facebook.com/hiphopdx
http://instagram.com/hiphopdx
For over 17 years, HipHopDX has been at the forefront of Hip Hop culture online, featuring over 2.7 million readers per month. As one of the longest-standing Hip Hop websites, DX not only stays current on Hip Hop culture, but continues to influence it, encourage it, and simultaneously reflect on its past. Our insightful, honest editorials, unbiased reviews, premier audio and video sections, and original video content, draws one of the most loyal followings online. Check us out at hiphopdx.com
-~-~~-~~~-~~-~-
Please watch: "Hopsin Talks End Of Funk Volume, Immaturity & Undercover Prodigy | SoulfulSundays"
https://www.youtube.com/watch?v=3V8Ds3q2tZA
-~-~~-~~~-~~-~-

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

12:07

The Secret Behind Numbers 3, 6, 9 Tesla Code Is Finally REVEALED!

The Secret Behind Numbers 3, 6, 9 Tesla Code Is Finally REVEALED!

The Secret Behind Numbers 3, 6, 9 Tesla Code Is Finally REVEALED!

SecretBehindThe Numbers 3, 6, 9 Tesla code Is Finally REVEALED!
Nikola Tesla did countless mysterious experiments, but he was a whole other mystery on his own. Almost all genius minds have a certain obsession. Nikola Tesla had a pretty big one!
He was walking around a block repeatedly for three times before entering a building, he would clean his plates with 18 napkins, he lived in hotel rooms only with a number divisible by 3. He would make calculations about things in his immediate environment to make sure the result is divisible by 3 and base his choices upon the results. He would do everything in sets of 3.
Some say he had OCD, some say he was very superstitious.
However, the truth is a lot deeper.
“If you knew the magnificence of the three, six and nine, you would have a key to the universe.” – Nikola Tesla
Music By:
1. FrostWaltz Alternate - Kevin Macleod
2. BluePaint Atlantean Twilight - Kevin Macleod
3. SoloCelloPassion - Doug Maxwell
https://youtu.be/LOJ50EUbWzg
UniversalTruth - http://www.369universe.com
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'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

Minhyong Kim: Connecting Number Theory to Physics

Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics. For more on Kim's work, read the full interview on Quanta Magazine's website: https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/
Video by Tom Medwell for Quanta Magazine.
https://www.quantamagazine.org/
Facebook: https://www.facebook.com/QuantaNews
Twitter: https://twitter.com/QuantaMagazine
Sign up for our weekly newsletter: http://eepurl.com/6FnWj
Quanta Magazine is an editorially independent publication launched by the Simons Foundation.

4:44

Introduction to Number Theory

Introduction to Number Theory

Introduction to Number Theory

Here we give a brief introduction to the branch of math known as number theory. This is a Bullis Student Tutors video -- made by students for students.
YouTube Channel: https://www.youtube.com/user/bullisstudenttutors
Google Mail: bullisstudenttutoring@gmail.com

Number Theory: Fermat's Little Theorem

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number TheoryPlaylist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
http://www.youtube.com/subscription_center?add_user=SocraticaStudios
Subject: Number Theory...

published: 12 Jan 2012

An Introduction to Number Theory : College Math

Subscribe Now:
http://www.youtube.com/subscription_center?add_user=Ehow
Watch More:
http://www.youtube.com/Ehow
Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

Philosophy of Numbers - Numberphile

We revisit the philosophy department and the question of whether numbers really exist?
FeaturingMarkJago from the University of Nottingham.
More links & stuff in full description below ↓↓↓
Earlier video on numbers' existence: https://youtu.be/1EGDCh75SpQ
Infinity paradoxes: https://youtu.be/dDl7g_2x74Q
Film and interview by Brady HaranEdit and animation: Pete McPartlan
Pete: https://twitter.com/petemcpartlan
Support us on Patreon: http://www.patreon.com/numberphile
NUMBERPHILE
Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile
Subscribe: http://bit.ly/Numberphile_Sub
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
Videos...

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Computer Science on Khan Academy: Learn select topics from computer science - algorithms (how we solve common problems in computer science and measure the efficiency of our solutions), cryptography (how we protect secret information), and information theory (how we encode and compress information).
About Khan A...

published: 29 Apr 2014

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foun...

published: 19 May 2014

The Magic Number Theory

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
Subscribe to HipHopDX on Youtube: http://bit.ly/dxsubscribe
Produced, Written & Hosted By: Murs
instagram.com/murs316
twitter.com/murs
twitch.tv/murs
Produced, Shot & Edited By: James Kreisberg
instagram.com/rolltheclipjames
Join the discussion on all socials #DXBreakdown
Check out more of DX here:
http://www.hiphopdx.com
https://twitter.com/hiphopdx
https://www.facebook.com/hiphopdx
http://instagram.com/hiphopdx
For over 17 years, HipHopDX has been at the fore...

For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

published: 30 Jan 2014

The Secret Behind Numbers 3, 6, 9 Tesla Code Is Finally REVEALED!

SecretBehindThe Numbers 3, 6, 9 Tesla code Is Finally REVEALED!
Nikola Tesla did countless mysterious experiments, but he was a whole other mystery on his own. Almost all genius minds have a certain obsession. Nikola Tesla had a pretty big one!
He was walking around a block repeatedly for three times before entering a building, he would clean his plates with 18 napkins, he lived in hotel rooms only with a number divisible by 3. He would make calculations about things in his immediate environment to make sure the result is divisible by 3 and base his choices upon the results. He would do everything in sets of 3.
Some say he had OCD, some say he was very superstitious.
However, the truth is a lot deeper.
“If you knew the magnificence of the three, six and nine, you would have a key to t...

'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

Minhyong Kim: Connecting Number Theory to Physics

Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics. For more on Kim's work, read the full interview on Quanta Magazine's website: https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/
Video by Tom Medwell for Quanta Magazine.
https://www.quantamagazine.org/
Facebook: https://www.facebook.com/QuantaNews
Twitter: https://twitter.com/QuantaMagazine
Sign up for our weekly newsletter: http://eepurl.com/6FnWj
Quanta Magazine is an editorially independent publication launched by the Simons Foundation.

published: 01 Dec 2017

Introduction to Number Theory

Here we give a brief introduction to the branch of math known as number theory. This is a Bullis Student Tutors video -- made by students for students.
YouTube Channel: https://www.youtube.com/user/bullisstudenttutors
Google Mail: bullisstudenttutoring@gmail.com

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number TheoryPlaylist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
http://www.youtube.com/subscription_center?add_user=SocraticaStudios
Subject: Number Theory
Teacher: Michael HarrisonArtist: Katrina de Dios

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number TheoryPlaylist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
http://www.youtube.com/subscription_center?add_user=SocraticaStudios
Subject: Number Theory
Teacher: Michael HarrisonArtist: Katrina de Dios

Subscribe Now:
http://www.youtube.com/subscription_center?add_user=Ehow
Watch More:
http://www.youtube.com/Ehow
Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

Subscribe Now:
http://www.youtube.com/subscription_center?add_user=Ehow
Watch More:
http://www.youtube.com/Ehow
Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory...

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Computer Science on Khan Academy: Learn select topics from computer science - algorithms (how we solve common problems in computer science and measure the efficiency of our solutions), cryptography (how we protect secret information), and information theory (how we encode and compress information).
About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Computer Science channel: https://www.youtube.com/channel/UC8uHgAVBOy5h1fDsjQghWCw?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Computer Science on Khan Academy: Learn select topics from computer science - algorithms (how we solve common problems in computer science and measure the efficiency of our solutions), cryptography (how we protect secret information), and information theory (how we encode and compress information).
About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Computer Science channel: https://www.youtube.com/channel/UC8uHgAVBOy5h1fDsjQghWCw?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an a...

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
Subscribe to HipHopDX on Youtube: http://bit.ly/dxsubscribe
Produced, Written & Hosted By: Murs
instagram.com/murs316
twitter.com/murs
twitch.tv/murs
Produced, Shot & Edited By: James Kreisberg
instagram.com/rolltheclipjames
Join the discussion on all socials #DXBreakdown
Check out more of DX here:
http://www.hiphopdx.com
https://twitter.com/hiphopdx
https://www.facebook.com/hiphopdx
http://instagram.com/hiphopdx
For over 17 years, HipHopDX has been at the forefront of Hip Hop culture online, featuring over 2.7 million readers per month. As one of the longest-standing Hip Hop websites, DX not only stays current on Hip Hop culture, but continues to influence it, encourage it, and simultaneously reflect on its past. Our insightful, honest editorials, unbiased reviews, premier audio and video sections, and original video content, draws one of the most loyal followings online. Check us out at hiphopdx.com
-~-~~-~~~-~~-~-
Please watch: "Hopsin Talks End Of Funk Volume, Immaturity & Undercover Prodigy | SoulfulSundays"
https://www.youtube.com/watch?v=3V8Ds3q2tZA
-~-~~-~~~-~~-~-

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
Subscribe to HipHopDX on Youtube: http://bit.ly/dxsubscribe
Produced, Written & Hosted By: Murs
instagram.com/murs316
twitter.com/murs
twitch.tv/murs
Produced, Shot & Edited By: James Kreisberg
instagram.com/rolltheclipjames
Join the discussion on all socials #DXBreakdown
Check out more of DX here:
http://www.hiphopdx.com
https://twitter.com/hiphopdx
https://www.facebook.com/hiphopdx
http://instagram.com/hiphopdx
For over 17 years, HipHopDX has been at the forefront of Hip Hop culture online, featuring over 2.7 million readers per month. As one of the longest-standing Hip Hop websites, DX not only stays current on Hip Hop culture, but continues to influence it, encourage it, and simultaneously reflect on its past. Our insightful, honest editorials, unbiased reviews, premier audio and video sections, and original video content, draws one of the most loyal followings online. Check us out at hiphopdx.com
-~-~~-~~~-~~-~-
Please watch: "Hopsin Talks End Of Funk Volume, Immaturity & Undercover Prodigy | SoulfulSundays"
https://www.youtube.com/watch?v=3V8Ds3q2tZA
-~-~~-~~~-~~-~-

SecretBehindThe Numbers 3, 6, 9 Tesla code Is Finally REVEALED!
Nikola Tesla did countless mysterious experiments, but he was a whole other mystery on his own. Almost all genius minds have a certain obsession. Nikola Tesla had a pretty big one!
He was walking around a block repeatedly for three times before entering a building, he would clean his plates with 18 napkins, he lived in hotel rooms only with a number divisible by 3. He would make calculations about things in his immediate environment to make sure the result is divisible by 3 and base his choices upon the results. He would do everything in sets of 3.
Some say he had OCD, some say he was very superstitious.
However, the truth is a lot deeper.
“If you knew the magnificence of the three, six and nine, you would have a key to the universe.” – Nikola Tesla
Music By:
1. FrostWaltz Alternate - Kevin Macleod
2. BluePaint Atlantean Twilight - Kevin Macleod
3. SoloCelloPassion - Doug Maxwell
https://youtu.be/LOJ50EUbWzg
UniversalTruth - http://www.369universe.com
If you wish to make a small donation, it would be Greatly appreciated - Thank you :)
PayPal Donation Link: https://www.paypal.com/cgi-bin/webscr?cmd=_s-xclick&hosted_button_id=6QHHNJVALEQDN

SecretBehindThe Numbers 3, 6, 9 Tesla code Is Finally REVEALED!
Nikola Tesla did countless mysterious experiments, but he was a whole other mystery on his own. Almost all genius minds have a certain obsession. Nikola Tesla had a pretty big one!
He was walking around a block repeatedly for three times before entering a building, he would clean his plates with 18 napkins, he lived in hotel rooms only with a number divisible by 3. He would make calculations about things in his immediate environment to make sure the result is divisible by 3 and base his choices upon the results. He would do everything in sets of 3.
Some say he had OCD, some say he was very superstitious.
However, the truth is a lot deeper.
“If you knew the magnificence of the three, six and nine, you would have a key to the universe.” – Nikola Tesla
Music By:
1. FrostWaltz Alternate - Kevin Macleod
2. BluePaint Atlantean Twilight - Kevin Macleod
3. SoloCelloPassion - Doug Maxwell
https://youtu.be/LOJ50EUbWzg
UniversalTruth - http://www.369universe.com
If you wish to make a small donation, it would be Greatly appreciated - Thank you :)
PayPal Donation Link: https://www.paypal.com/cgi-bin/webscr?cmd=_s-xclick&hosted_button_id=6QHHNJVALEQDN

'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 1...

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

Minhyong Kim: Connecting Number Theory to Physics

Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics. For more on Kim's work, re...

Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics. For more on Kim's work, read the full interview on Quanta Magazine's website: https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/
Video by Tom Medwell for Quanta Magazine.
https://www.quantamagazine.org/
Facebook: https://www.facebook.com/QuantaNews
Twitter: https://twitter.com/QuantaMagazine
Sign up for our weekly newsletter: http://eepurl.com/6FnWj
Quanta Magazine is an editorially independent publication launched by the Simons Foundation.

Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics. For more on Kim's work, read the full interview on Quanta Magazine's website: https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/
Video by Tom Medwell for Quanta Magazine.
https://www.quantamagazine.org/
Facebook: https://www.facebook.com/QuantaNews
Twitter: https://twitter.com/QuantaMagazine
Sign up for our weekly newsletter: http://eepurl.com/6FnWj
Quanta Magazine is an editorially independent publication launched by the Simons Foundation.

Introduction to Number Theory

Here we give a brief introduction to the branch of math known as number theory. This is a Bullis Student Tutors video -- made by students for students.
YouTub...

Here we give a brief introduction to the branch of math known as number theory. This is a Bullis Student Tutors video -- made by students for students.
YouTube Channel: https://www.youtube.com/user/bullisstudenttutors
Google Mail: bullisstudenttutoring@gmail.com

Here we give a brief introduction to the branch of math known as number theory. This is a Bullis Student Tutors video -- made by students for students.
YouTube Channel: https://www.youtube.com/user/bullisstudenttutors
Google Mail: bullisstudenttutoring@gmail.com

Number theory and its applications by Dr. Kotyada Srinivas

published: 17 Oct 2014

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foun...

'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

published: 01 Jul 2015

Pi hiding in prime regularities

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Check out Remix careers: https://www.remix.com/jobs
The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2
Special thanks to the following patrons: http://3b1b.co/leibniz-thanks
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various s...

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

A crash course in Algebraic Number Theory

A quick proof of the PrimeIdealTheorem (algebraic analog of the Prime Number Theorem) is presented.
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
This is an excerpt of lecture 6 in the series "Introduction to Analytic Number Theory" by Ram Murty. Full playlist of this mini course is available here: https://www.youtube.com/playlist?list=PLhsb6tmzSpiwLds3DD62o1MvI2MS2bby7

Number Theory

Progress in Prime Number Theory (Roger Heath-Brown)

Abstract: This lecture will discuss prime numbers and their history, along with some of the many open problems concerning them. There has been much exciting progress over the past few years, and the lecture will provide an overview of what has been achieved, and where the current areas of activity lie.

Introduction to Number Theory

The Queen of Mathematics - Professor Raymond Flood

Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." The properties of primes play a crucial part in number theory. An intriguing question is how they are distributed among the other integers. The 19th century saw progress in answering this question with the proof of the Prime Number Theorem although it also saw Bernhard Riemann posing what many think to be the greatest unsolved problem in mathematics - the Rieman Hypothesis.
The transcript and downloadable versions of the lecture are available from the Gresham College website:
http://www.gresham.ac.uk/lectures-and-events/the-queen-of-mathematics
Gresham College has been giving free public lectures since 1597. This tra...

published: 31 Jan 2013

MathHistory13: The number theory revival

After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
My research papers can be found at my ResearchGate page, at https://www.researchga...

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an a...

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 1...

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

Pi hiding in prime regularities

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Check out Remix careers: https://www.remix.com/jobs
The fact th...

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Check out Remix careers: https://www.remix.com/jobs
The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2
Special thanks to the following patrons: http://3b1b.co/leibniz-thanks
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3Blue1Brown
Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown
Reddit: https://www.reddit.com/r/3Blue1Brown

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Check out Remix careers: https://www.remix.com/jobs
The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2
Special thanks to the following patrons: http://3b1b.co/leibniz-thanks
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3Blue1Brown
Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown
Reddit: https://www.reddit.com/r/3Blue1Brown

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

A quick proof of the PrimeIdealTheorem (algebraic analog of the Prime Number Theorem) is presented.
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
This is an excerpt of lecture 6 in the series "Introduction to Analytic Number Theory" by Ram Murty. Full playlist of this mini course is available here: https://www.youtube.com/playlist?list=PLhsb6tmzSpiwLds3DD62o1MvI2MS2bby7

A quick proof of the PrimeIdealTheorem (algebraic analog of the Prime Number Theorem) is presented.
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
This is an excerpt of lecture 6 in the series "Introduction to Analytic Number Theory" by Ram Murty. Full playlist of this mini course is available here: https://www.youtube.com/playlist?list=PLhsb6tmzSpiwLds3DD62o1MvI2MS2bby7

Progress in Prime Number Theory (Roger Heath-Brown)

Abstract: This lecture will discuss prime numbers and their history, along with some of the many open problems concerning them. There has been much exciting pr...

Abstract: This lecture will discuss prime numbers and their history, along with some of the many open problems concerning them. There has been much exciting progress over the past few years, and the lecture will provide an overview of what has been achieved, and where the current areas of activity lie.

Abstract: This lecture will discuss prime numbers and their history, along with some of the many open problems concerning them. There has been much exciting progress over the past few years, and the lecture will provide an overview of what has been achieved, and where the current areas of activity lie.

Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." The properties of primes play a crucial part in number theory. An intriguing question is how they are distributed among the other integers. The 19th century saw progress in answering this question with the proof of the Prime Number Theorem although it also saw Bernhard Riemann posing what many think to be the greatest unsolved problem in mathematics - the Rieman Hypothesis.
The transcript and downloadable versions of the lecture are available from the Gresham College website:
http://www.gresham.ac.uk/lectures-and-events/the-queen-of-mathematics
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently nearly 1,500 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: http://www.facebook.com/pages/Gresham-College/14011689941

Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." The properties of primes play a crucial part in number theory. An intriguing question is how they are distributed among the other integers. The 19th century saw progress in answering this question with the proof of the Prime Number Theorem although it also saw Bernhard Riemann posing what many think to be the greatest unsolved problem in mathematics - the Rieman Hypothesis.
The transcript and downloadable versions of the lecture are available from the Gresham College website:
http://www.gresham.ac.uk/lectures-and-events/the-queen-of-mathematics
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently nearly 1,500 lectures free to access or download from the website.
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MathHistory13: The number theory revival

After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connec...

After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Number Theory: Fermat's Little Theorem

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
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Subject: Number Theory
Teacher: Michael HarrisonArtist: Katrina de Dios

An Introduction to Number Theory : College Math

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Number theory is actually a pretty intensive course that's in junior or senior levels of undergraduate college mathematics. Get an introduction to number theory with help from a longtime mathematics educator in this free video clip.
Expert: JimmyChangFilmmaker: Christopher Rokosz
SeriesDescription: Topics like number theory will start to come into play as your mathematics career advances towards the college level and beyond. Learn about the ins and outs of college math with help from a longtime mathematics educator in this free video series.

How can we estimate the number of primes up to x?
Watch the next lesson: https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/time-space-tradeoff?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
Missed the previous lesson? https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/trial-division-primality-test-using-a-sieve-prime-adventure-part-5?utm_source=YT&utm_medium=Desc&utm_campaign=computerscience
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48:27

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbe...

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

13:00

The Magic Number Theory

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they...

The Magic Number Theory

What do Beastie Boys, The Fugees, Migos and Flatbush Zombies all have in common? Yes, they all rap - but the second most obvious answer is that all these groups have 3 MC’s. Of all possible MC configurations, the trio has been the most impactful throughout Hip Hop history. The Magic NumberTheory: Let’s break it down…
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The Secret Behind Numbers 3, 6, 9 Tesla Code Is Finally REVEALED!

SecretBehindThe Numbers 3, 6, 9 Tesla code Is Finally REVEALED!
Nikola Tesla did countless mysterious experiments, but he was a whole other mystery on his own. Almost all genius minds have a certain obsession. Nikola Tesla had a pretty big one!
He was walking around a block repeatedly for three times before entering a building, he would clean his plates with 18 napkins, he lived in hotel rooms only with a number divisible by 3. He would make calculations about things in his immediate environment to make sure the result is divisible by 3 and base his choices upon the results. He would do everything in sets of 3.
Some say he had OCD, some say he was very superstitious.
However, the truth is a lot deeper.
“If you knew the magnificence of the three, six and nine, you would have a key to the universe.” – Nikola Tesla
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The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

There's a ludicrous amount of fan theories floating about the wider world of Harry Potter... It takes a lot to be the most ridiculous when every theory creates a new high, or low, for crazy conceptions involving the Potter crew....

Feeling a little confused by this new season of Westworld? You’re not alone. Luckily, the show’s own Jimmi Simpson is here to weigh in on a whole bunch of insane fan theories and let you know how close you are to figuring it all out. Sort of. As with all these “Actors Read Fan Theories” videos from Elle, Simpson is…. Read more... ....

(CNN)PresidentDonald Trump has turned an unsubstantiated rumor that the FBI secretly placed an informant in his 2016 campaign into a full-blown conspiracy theory in the last five days ... This is not the first time Trump has propelled a conspiracy theory with no evidence ... "He frequently lies and has a long and well-documented career engaging in conspiracy theories about all manner of subjects, with no concrete evidence provided."....

Search efforts to find the missing Malaysia AirlinesFlight 370 that mysteriously vanished more than four years ago will finally end next week, leaving the world with only theories — for now — as to what happened to the aircraft ... ....

MathHistory22: Algebraic number theory and rings I

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

This talk was originally given to an audience of school students aged 16-17 as part of a mathematics enrichment event at the University of Cambridge. Recorded 19 June 2015.
Prime numbers are fundamentally important in mathematics. Discover some of the beautiful properties of prime numbers in this talk by Dr Vicky Neale, and learn about some of the unsolved problems in number theory that mathematicians are working on today.

30:42

Pi hiding in prime regularities

A story of pi, prime numbers, and complex numbers, and how number theory braids them toget...

Pi hiding in prime regularities

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
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The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2
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Speaker(s): Jordan Ellenberg (University of Wisconsin-Madison)
Location: MSRI: Simons AuditoriumFebruary 06, 2017
Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmetic-statistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
Video taken from:
https://www.msri.org/workshops/801/schedules/21765

55:48

LMS Popular Lecture Series 2013, Addictive Number Theory

Addictive Number Theory by Dr Vicky Neale
Held at the Institute of Education in London

A crash course in Algebraic Number Theory

A quick proof of the PrimeIdealTheorem (algebraic analog of the Prime Number Theorem) is presented.
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
This is an excerpt of lecture 6 in the series "Introduction to Analytic Number Theory" by Ram Murty. Full playlist of this mini course is available here: https://www.youtube.com/playlist?list=PLhsb6tmzSpiwLds3DD62o1MvI2MS2bby7

1:11:44

Andrew Wiles’ historical Fermat's Last Theorem talk at the 1998 ICM

Twenty Years of Number Theory
Andrew Wiles
Princeton University
ICM Berlin 19.08.1998
ht...

MathHistory13: The number theory revival...

Terence Tao - Recent progress in additive prime nu...

Number Three

There's only two songs in me and I just wrote the thirdDon't know where I got the inspiration or how I wrote the wordsSpent my whole life just digging up my music's shallow graveFor the two songs in me and the third one I just madeA rich man once told me";Hey life's a funny thing";A poor man once told meThat he can't afford to speakNow I'm in the middle like a bird without a beak 'causeThere's just two songs in me and I just wrote the thirdDon't know where I got the inspiration or how I wrote the wordsSpent my whole life just digging up my music's shallow graveFor the two songs in me and the third one I just madeSo I went to the PresidentAnd I asked old what's-his-nameHas he ever gotten writer's blockOr something like the sameHe just started talkingLike he was on TV";If there's just two songs in ya, boyWhaddaya want from me?";So I bought myself some denim pantsAnd a silver guitarBut I politely told the ladies";You'll still have to call me SirBecause I have to keep my self-respectI'll never be a starSince there's just two songs in meAnd this is Number Three";

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There's a ludicrous amount of fan theories floating about the wider world of Harry Potter... It takes a lot to be the most ridiculous when every theory creates a new high, or low, for crazy conceptions involving the Potter crew....

Feeling a little confused by this new season of Westworld? You’re not alone. Luckily, the show’s own Jimmi Simpson is here to weigh in on a whole bunch of insane fan theories and let you know how close you are to figuring it all out. Sort of. As with all these “Actors Read Fan Theories” videos from Elle, Simpson is…. Read more... ....

(CNN)PresidentDonald Trump has turned an unsubstantiated rumor that the FBI secretly placed an informant in his 2016 campaign into a full-blown conspiracy theory in the last five days ... This is not the first time Trump has propelled a conspiracy theory with no evidence ... "He frequently lies and has a long and well-documented career engaging in conspiracy theories about all manner of subjects, with no concrete evidence provided."....

Search efforts to find the missing Malaysia AirlinesFlight 370 that mysteriously vanished more than four years ago will finally end next week, leaving the world with only theories — for now — as to what happened to the aircraft ... ....

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